is well established, in which \(\mathbf {H}_{\mathrm {eff}}\)  is an effective
magnetic field and \(\alpha _{G}\)  the Gilbert damping constant. \(\mathbf {H}_{\mathrm {eff}}\)  contains a stochastic term with a correlation function that
in equilibrium obeys the fluctuation-dissipation theorem in terms of the
Gilbert damping and temperature, and can be used to compute the spin-spin
correlation and response functions [1]}. The linearized
LLG equation, equivalent to the Heisenberg equation to lowest order in the
Holstein-Primakoff expansion [2]}, is appropriate for weakly or
thermally excited magnon gases. Its solutions are the magnon dispersion
relations, amplitudes, and group velocities, which are the starting point for
transport theories such as the Boltzmann formalism. The magnetization dynamics
consists of small transverse fluctuations \(\mathbf {m}_{\bot }\)  that reduce the
net magnetization magnetization \(M\)  relative to the ground state \(M_{0}\)
\(-m_{z}=\frac{\mathbf {m}_{\bot }^{2}}{2}. \) 
